The Reds had taken the lead thanks to Emre Can’s deflected shot in the ninth minute, but Chelsea soon fought back through Gary Cahill’s equaliser.
The Blues then took a 2-1 lead when Diego Costa fired home in the 67th minute.
But with Liverpool pressing for an equaliser, Steven Gerrard’s shot appeared to hit the hand of Cahill in the penalty area.
And despite loud protests from the Kop, referee Taylor waved play-on.
“We worked so hard in the game but we need those decisions that are obvious, it was a clear, clear penalty,” said Rodgers.
“I was very disappointed because the players gave everything in the game. We deserved at least a point for sure.
“You need that wee bit of luck going for you and it isn’t at the moment. It has been a tough week.”
Rodgers, who watched his side slump to a third straight defeat in all competitions, added: “We have to keep working hard and hope get some of our injured players back in.
“There is no magic formula but I have a brilliant group of players here. We will be ready and firing after the international break.”
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